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Properties

Label 605.4.a.c.1.1
Level $605$
Weight $4$
Character 605.1
Self dual yes
Analytic conductor $35.696$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [605,4,Mod(1,605)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("605.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(605, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 605.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,1,-5,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(35.6961555535\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 605.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -5.00000 q^{3} -7.00000 q^{4} +5.00000 q^{5} -5.00000 q^{6} +29.0000 q^{7} -15.0000 q^{8} -2.00000 q^{9} +5.00000 q^{10} +35.0000 q^{12} -88.0000 q^{13} +29.0000 q^{14} -25.0000 q^{15} +41.0000 q^{16} +21.0000 q^{17} -2.00000 q^{18} +105.000 q^{19} -35.0000 q^{20} -145.000 q^{21} +160.000 q^{23} +75.0000 q^{24} +25.0000 q^{25} -88.0000 q^{26} +145.000 q^{27} -203.000 q^{28} -165.000 q^{29} -25.0000 q^{30} -85.0000 q^{31} +161.000 q^{32} +21.0000 q^{34} +145.000 q^{35} +14.0000 q^{36} -15.0000 q^{37} +105.000 q^{38} +440.000 q^{39} -75.0000 q^{40} -270.000 q^{41} -145.000 q^{42} +12.0000 q^{43} -10.0000 q^{45} +160.000 q^{46} -370.000 q^{47} -205.000 q^{48} +498.000 q^{49} +25.0000 q^{50} -105.000 q^{51} +616.000 q^{52} -615.000 q^{53} +145.000 q^{54} -435.000 q^{56} -525.000 q^{57} -165.000 q^{58} +396.000 q^{59} +175.000 q^{60} -835.000 q^{61} -85.0000 q^{62} -58.0000 q^{63} -167.000 q^{64} -440.000 q^{65} -540.000 q^{67} -147.000 q^{68} -800.000 q^{69} +145.000 q^{70} -187.000 q^{71} +30.0000 q^{72} -58.0000 q^{73} -15.0000 q^{74} -125.000 q^{75} -735.000 q^{76} +440.000 q^{78} -620.000 q^{79} +205.000 q^{80} -671.000 q^{81} -270.000 q^{82} +828.000 q^{83} +1015.00 q^{84} +105.000 q^{85} +12.0000 q^{86} +825.000 q^{87} -1535.00 q^{89} -10.0000 q^{90} -2552.00 q^{91} -1120.00 q^{92} +425.000 q^{93} -370.000 q^{94} +525.000 q^{95} -805.000 q^{96} -90.0000 q^{97} +498.000 q^{98} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.353553 0.176777 − 0.984251i \(-0.443433\pi\)
0.176777 + 0.984251i \(0.443433\pi\)
\(3\) −5.00000 −0.962250 −0.481125 − 0.876652i \(-0.659772\pi\)
−0.481125 + 0.876652i \(0.659772\pi\)
\(4\) −7.00000 −0.875000
\(5\) 5.00000 0.447214
\(6\) −5.00000 −0.340207
\(7\) 29.0000 1.56585 0.782926 − 0.622114i \(-0.213726\pi\)
0.782926 + 0.622114i \(0.213726\pi\)
\(8\) −15.0000 −0.662913
\(9\) −2.00000 −0.0740741
\(10\) 5.00000 0.158114
\(11\) 0 0
\(12\) 35.0000 0.841969
\(13\) −88.0000 −1.87745 −0.938723 − 0.344671i \(-0.887990\pi\)
−0.938723 + 0.344671i \(0.887990\pi\)
\(14\) 29.0000 0.553613
\(15\) −25.0000 −0.430331
\(16\) 41.0000 0.640625
\(17\) 21.0000 0.299603 0.149801 − 0.988716i \(-0.452137\pi\)
0.149801 + 0.988716i \(0.452137\pi\)
\(18\) −2.00000 −0.0261891
\(19\) 105.000 1.26782 0.633912 − 0.773405i \(-0.281448\pi\)
0.633912 + 0.773405i \(0.281448\pi\)
\(20\) −35.0000 −0.391312
\(21\) −145.000 −1.50674
\(22\) 0 0
\(23\) 160.000 1.45054 0.725268 − 0.688467i \(-0.241716\pi\)
0.725268 + 0.688467i \(0.241716\pi\)
\(24\) 75.0000 0.637888
\(25\) 25.0000 0.200000
\(26\) −88.0000 −0.663778
\(27\) 145.000 1.03353
\(28\) −203.000 −1.37012
\(29\) −165.000 −1.05654 −0.528271 − 0.849076i \(-0.677160\pi\)
−0.528271 + 0.849076i \(0.677160\pi\)
\(30\) −25.0000 −0.152145
\(31\) −85.0000 −0.492466 −0.246233 − 0.969211i \(-0.579193\pi\)
−0.246233 + 0.969211i \(0.579193\pi\)
\(32\) 161.000 0.889408
\(33\) 0 0
\(34\) 21.0000 0.105926
\(35\) 145.000 0.700271
\(36\) 14.0000 0.0648148
\(37\) −15.0000 −0.0666482 −0.0333241 − 0.999445i \(-0.510609\pi\)
−0.0333241 + 0.999445i \(0.510609\pi\)
\(38\) 105.000 0.448243
\(39\) 440.000 1.80657
\(40\) −75.0000 −0.296464
\(41\) −270.000 −1.02846 −0.514231 − 0.857652i \(-0.671922\pi\)
−0.514231 + 0.857652i \(0.671922\pi\)
\(42\) −145.000 −0.532714
\(43\) 12.0000 0.0425577 0.0212789 − 0.999774i \(-0.493226\pi\)
0.0212789 + 0.999774i \(0.493226\pi\)
\(44\) 0 0
\(45\) −10.0000 −0.0331269
\(46\) 160.000 0.512842
\(47\) −370.000 −1.14830 −0.574149 − 0.818751i \(-0.694667\pi\)
−0.574149 + 0.818751i \(0.694667\pi\)
\(48\) −205.000 −0.616442
\(49\) 498.000 1.45190
\(50\) 25.0000 0.0707107
\(51\) −105.000 −0.288293
\(52\) 616.000 1.64277
\(53\) −615.000 −1.59390 −0.796950 − 0.604045i \(-0.793555\pi\)
−0.796950 + 0.604045i \(0.793555\pi\)
\(54\) 145.000 0.365407
\(55\) 0 0
\(56\) −435.000 −1.03802
\(57\) −525.000 −1.21996
\(58\) −165.000 −0.373544
\(59\) 396.000 0.873810 0.436905 − 0.899508i \(-0.356075\pi\)
0.436905 + 0.899508i \(0.356075\pi\)
\(60\) 175.000 0.376540
\(61\) −835.000 −1.75264 −0.876318 − 0.481733i \(-0.840007\pi\)
−0.876318 + 0.481733i \(0.840007\pi\)
\(62\) −85.0000 −0.174113
\(63\) −58.0000 −0.115989
\(64\) −167.000 −0.326172
\(65\) −440.000 −0.839620
\(66\) 0 0
\(67\) −540.000 −0.984649 −0.492325 − 0.870412i \(-0.663853\pi\)
−0.492325 + 0.870412i \(0.663853\pi\)
\(68\) −147.000 −0.262152
\(69\) −800.000 −1.39578
\(70\) 145.000 0.247583
\(71\) −187.000 −0.312575 −0.156287 − 0.987712i \(-0.549953\pi\)
−0.156287 + 0.987712i \(0.549953\pi\)
\(72\) 30.0000 0.0491046
\(73\) −58.0000 −0.0929916 −0.0464958 − 0.998918i \(-0.514805\pi\)
−0.0464958 + 0.998918i \(0.514805\pi\)
\(74\) −15.0000 −0.0235637
\(75\) −125.000 −0.192450
\(76\) −735.000 −1.10935
\(77\) 0 0
\(78\) 440.000 0.638720
\(79\) −620.000 −0.882980 −0.441490 − 0.897266i \(-0.645550\pi\)
−0.441490 + 0.897266i \(0.645550\pi\)
\(80\) 205.000 0.286496
\(81\) −671.000 −0.920439
\(82\) −270.000 −0.363616
\(83\) 828.000 1.09500 0.547499 − 0.836806i \(-0.315580\pi\)
0.547499 + 0.836806i \(0.315580\pi\)
\(84\) 1015.00 1.31840
\(85\) 105.000 0.133986
\(86\) 12.0000 0.0150464
\(87\) 825.000 1.01666
\(88\) 0 0
\(89\) −1535.00 −1.82820 −0.914099 − 0.405490i \(-0.867101\pi\)
−0.914099 + 0.405490i \(0.867101\pi\)
\(90\) −10.0000 −0.0117121
\(91\) −2552.00 −2.93981
\(92\) −1120.00 −1.26922
\(93\) 425.000 0.473876
\(94\) −370.000 −0.405985
\(95\) 525.000 0.566988
\(96\) −805.000 −0.855833
\(97\) −90.0000 −0.0942074 −0.0471037 − 0.998890i \(-0.514999\pi\)
−0.0471037 + 0.998890i \(0.514999\pi\)
\(98\) 498.000 0.513322
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 605.4.a.c.1.1 yes 1
11.10 odd 2 605.4.a.a.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
605.4.a.a.1.1 ✓ 1 11.10 odd 2
605.4.a.c.1.1 yes 1 1.1 even 1 trivial